Introduction To Modern Planar Transmission Lines. Anand K. Verma

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rel="nofollow" href="#ulink_9c7233f6-43ab-558b-9aa9-b87bd85fc10e">Figure 5.7 Electrical grouping of the materials media in the (μ_r, ε_r)‐plane.

Following the above‐discussed nomenclature scheme of a material medium, it is natural to envisage an artificial medium with both the permittivity and permeability as negative quantities. Such an engineered medium could be called a double negative medium (DNG) medium. It is shown in the third quadrant of Fig (5.7). It has a negative refractive index (−n). The DPS medium supports the wave propagation such that both the phase velocity v_p and group velocity v_g are in the same direction, whereas the DNG medium supports wave propagation with phase velocity v_p and group velocity v_g in opposite directions. Such EM‐wave is known as the backward wave. However, no DNG material is known to exist in nature. The DNG material medium has been synthesized over certain frequency bands, from the microwave to optical frequency ranges.

      The DNG materials are commonly known as metamaterials. However, artificially engineered ENG, MNG, ENZ, and MNZ media also belong to the metamaterials. Even the DPS could be engineered, and called a metamaterial, to realize the controlled values of permittivity and permeability to meet the specific design requirement. One such requirement is to develop the microwave absorbers discussed in subsection (5.5.8). Several dedicated books are available on the topics of the metamaterials [B.6–B.10].

      Depending on their characteristics, several other names have also been proposed for the metamaterials [B.7, B.10]. Veselago proposed the concept and electrodynamics of the metamaterials [J.3]. The metamaterials have been synthesized by embedding either the resonating type inclusions or the C‐L type artificial transmission line inclusions in a host medium. The metamaterials could be either resonant metamaterials or nonresonant metamaterials, so the metamaterials are structured composite materials. Starting with J.C. Bose, such composite materials have a long history [J.4–J.6]. The concept of composite materials and their modeling is discussed in section (6.3) of chapter 6. The C‐L line supports the backward wave propagation. It is discussed in section (3.4) of chapter 3. A metamaterial, i.e. the DNG medium, is also called the backward wave (BW) medium, or BW material [J.7]. The normal forward wave supporting DPS medium could be called forward wave (FW) medium, or FW material.

      The physical dimensions of the resonating inclusions, or the C‐L line inclusions, are much smaller than the operating wavelengths. The metamaterials could be treated as the homogenized medium described by the parameters permittivity and permeability. The inclusions are arranged in the host medium either periodically, or nonperiodically. The periodically arranged inclusions, with a period λ_g/2, form another group of metamaterials known as the electronic bandgap (EBG) materials. It is discussed in chapters 19 and 20. Even the ENG and MNG materials have been artificially developed using such inclusions.

The (μ_r, ε_r)‐plane of Fig (5.7) summarizes the above‐discussed four‐groups of material media with their basic electrical characteristics. The circuit models and further wave characteristics of these media are shown in Fig (5.10). The synthesis, modeling, and some illustrative applications of the metamaterials are discussed in chapter 21. This section only presents some basic electrodynamics properties of the DNG medium.

      

      5.5.2 EM‐Waves in DNG Medium

      This section considers the EM‐wave propagation in the lossless DNG medium, the concept of the negative refractive index, the basic circuit model of a DNG medium, and the lossy DNG medium.

      Lossless DNG Medium

      Maxwell's equation (4.4.11a,b) of chapter 4 could be written for a lossless (σ = 0) DNG medium by using (−|ε_r|) and (−|μ_r|) in place of their usual positive values:

      (5.5.2)

Equations (4.5.31a) and (4.2.3b) of chapter 4 show Maxwell's equations for a DPS medium in terms of the wavevector and the field vectors . They form the wavevector triplet () relations in the right‐hand (RH) coordinate system, shown in Fig (5.8a) for a DPS medium. Normally, the direction of the wavevector determines the direction of the wave propagation, i.e. the direction of the phase velocity v_p. However, the Poynting vector () given by equation (4.4.20) provides another power‐vector triplet () that determines the true direction of wave propagation. It is the direction of the energy flow from the source to a load. So the corresponding group velocity v_g defines the direction of the wave propagation.

      Figure (5.8c) shows both the sets of the vector triplets of a DPS medium. The field vectors are rotated in cyclic order so that the power flow is maintained from left to right‐hand side, i.e. from a source located at the origin O, power flows outwardly in a positive direction. For both the phase and group velocities, the DPS medium follows the right‐hand (RH) coordinate system, so a normal DPS medium is called the right‐handed, i.e. the RH‐medium. As the RH‐system holds for both triplets, they are combined into one diagram as shown in Fig (5.8c). For a DPS medium, the directions of the vectors and are identical, i.e. they are parallel vectors giving the relation . Therefore, in a DPS medium, both the phase and group velocities are in the same direction giving . It is shown in the first quadrant of Fig (5.7). The phase of the propagating EM‐wave in the DPS medium lags while traveling in the

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