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      The condition for the normal dispersion is dv_p/dβ < 0, dv_p/dλ > 0 and for the anomalous dispersion, it is dv_p/dβ > 0, dv_p/dλ < 0.

3.4 Linear Dispersive Transmission Lines

      The velocity of EM‐wave propagation in a dispersive medium has been discussed in the previous section. The transmission line is a 1D wave‐supporting medium. The lossy line is a dispersive medium, whereas a lossless line, modeled through the line constants L and C, is a nondispersive medium. It acts as a low‐pass filter (LPF). This section shows that a reactively loaded lossless line could be a dispersive medium. A variety of transmission line structures with interesting properties can be developed by using several additional combinations of C and L. Such line structures can be realized with the lumped elements and also by the modification of planar transmission lines. The transmission medium with negative relative permittivity and negative relative permeability has been synthesized with the help of the modified, i.e. reactively loaded line structures. These wave supporting media form a new class of materials known as the metamaterials. They do not exist in nature. However, these novel media have been developed with the defects in the transmission line and using the embedded resonators in the transmission line [B.19, J.8]. The present section considers only an infinitesimal section of the transmission line, modeled as a lumped circuit elements network. The modeled network is reactively loaded to get the loaded line. However, in practice, a finite length of the line is periodically loaded with reactance. Such reactively loaded lines offer novel and improved designs of microwave components and circuits. The periodically loaded line, creating the 1D – EBG and metalines are discussed in the chapters 19 and 22.

      3.4.1 Wave Equation of Dispersive Transmission Lines

Figure (3.27a) shows the shunt inductor loaded line structure. An infinitesimally small Δx line length is considered. The cascading of a large number line sections, under the condition Δx → 0, forms a continuous line. The standard Δx long host transmission line section is modeled by the distributed line constants L and C p.u.l. The host line section is loaded with the inclusion, i.e. with additional shunt inductance L_sh. The loading element, i.e. inclusion is shown in the gray box, so the periodic embedding of reactive inclusion in the host line medium creates an artificial 1D mixture medium. The concept of the mixture material is further discussed in subsection (6.3.1) of chapter 6.

The inclusion lumped shunt inductance L_sh distributed over the length Δx is (L_sh/Δx) p.u.l. The total series inductance, (L Δx), gives the total series impedance Z = j ωL Δx and a combination of shunt capacitance (C Δx) and shunt inductance (L_sh/Δx) gives the total shunt admittance, Y = j (ωCΔx − Δx/(ωL_sh)). The transmission line equations for the shunt inductor loaded line are obtained as follows:

      (3.4.1)

      For the limiting case, Δx → 0, Δt → 0, the following expressions are obtained:

(3.4.2)

      The voltage v across the shunt inductor is

(3.4.3)

From equation (3.4.2a,b), the following voltage wave equation is obtained:

      (3.4.4)

On substituting ∂i₂/∂t from equation (3.4.3) in the above equation, the above wave equation is reduced to

      (3.4.5)

      where is the phase velocity of the standard nondispersive line without shunt loaded inductance, i.e. for the case L_sh → ∞. For a line in the air medium, the phase velocity is the velocity of the EM‐wave in a vacuum, i.e. v₀ = c. The cut‐off frequency is defined as . Finally, the above voltage wave equation of the dispersive transmission line is rewritten as

(3.4.6)

Equation (3.4.6) is known as the Klein–Gordon equation [B.20]. For a line without cut‐off frequency, i.e. for ω_c = 0, the above equation is reduced to the standard dispersionless transmission line equation (2.1.24) of chapter 2. On assuming the harmonic solution for the forward‐moving voltage, the voltage on the loaded line is

(3.4.7)

On substituting equation (3.4.7) in equation (3.4.6), the following dispersion relation is obtained:

(3.4.8)

Equation (3.4.8) is identical to equation (3.3.8a) for the plasma medium. The cut‐off frequency ω_c of the loaded line corresponds to the plasma frequency ω_p. Thus, a plasma medium can be modeled by the shunt inductor loaded transmission line. Further, Fig (3.27a) shows that below the cut‐off frequency, i.e. for ω < ω_c the circuit is reduced to the L‐L circuit presenting inductors in both the series and shunt arm. Such a medium is called the epsilon negative (ENG) medium, discussed in section (5.5) of chapter 5.

      For frequency ω < ω_c, the propagation constant β is the imaginary quantity and the voltage wave cannot propagate on the shunt inductor loaded line. The voltage wave propagates only for ω > ω_c. Thus, the shunt inductor loaded line behaves like a high‐pass filter (HPF). It is like a metallic

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