Introduction To Modern Planar Transmission Lines. Anand K. Verma

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.

      Series Capacitor Loaded LC‐Line

The normal LC‐line can also be loaded with a series capacitor C_s 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

(3.4.24)

      where the phase velocity of the unloaded LC‐line is . The cut‐off frequency of the loaded line is . For ω > ω_c, the line behaves as the HPF. In absence of the series capacitance C_s loading, the line behaves like the LPF. The series‐arm impedance is capacitive at a frequency below cut‐off, i.e. for ω < ω_c. The circuit of Fig (3.30) is reduced to the C‐C line, i.e. a line with capacitive elements in both the series and shunt arms. It corresponds to the mu‐negative (MNG) medium discussed in the section (5.5) of chapter 5. The C‐C line blocks the low‐frequency signal. Therefore, for the frequency ω < ω_c, the propagation is in the evanescent mode with high attenuation. The phase and group velocities of the propagating waves are obtained as

(3.4.25)

Equations (3.4.24) and (3.4.25) for β, v_p, and v_g are identical to equations (3.4.10a) and (3.4.10b) for the transmission line shown in Fig (3.27a). Thus, Fig (3.27b and c) also show the behavior of the present series capacitor loaded LC‐line. This line supports the forward wave and it is a normal dispersive transmission line. It supports the fast‐wave above the cut‐off frequency. This line is the dual structure of the line shown in Fig (3.29) that supports the backward wave propagation.

      Additional numbers of configurations for the loaded transmission lines could be obtained. For instance, the L‐C section of a line, supporting the forward wave, could be cascaded with the C‐L section of a line, supporting the backward wave. The composite line forms an interesting kind of the transmission line structure [B.19, J.8]. Both the series and parallel reactive loading of the lines can be done. Such loaded line structures have been realized in the planar technology to obtain novel properties useful for the development of novel microwave devices. They form the so‐called metamaterials. The concept of metamaterials has been introduced in chapter 5 and elaborated in chapter 21. Chapter 22 considers the planar 1D‐metalines and 2D‐planar metasurfaces, and chapter 19 discusses the planar periodic transmission lines.

References

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