Introduction To Modern Planar Transmission Lines. Anand K. Verma

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(5.1.12b) has been also obtained in equation (5.1.5c) from the concept of reflectance and transmittance. The above relation shows that the power balance Γ² + τ² = 1 does not hold in the present case. The modified power balance relation is given by equation (5.1.12b). It shows that the transmission coefficient could be even more than unity, as it is multiplied by a factor However, the above‐given power balance relation must be maintained [B.1].

      5.1.2 The Interface of a Dielectric and Perfect Conductor

Medium #2 could be a conducting medium, with the intrinsic impedance, For a PEC medium, σ → ∞ , η₂ → 0.The PEC has infinite permittivity, i.e. ε → ∞. Using equation (5.1.3a), the interface provides the total reflection with the reflection coefficient Γ = − 1; i.e. a phase reversal for the reflected component. The total tangential electric field at the interface (x = 0⁻) is zero. There is no field in the medium #2. The total electric field, using an equation (5.1.6a), in medium #1 is,

(5.1.13)

The expression (5.1.13) shows the formation of a standing wave in the medium #1, without any traveling wave. The standing wave for the H_z field component can be easily obtained. At the PEC surface, the E_y‐field is zero, and H_z‐field is at maximum.

      The total reflection at the interface also occurs for η₂ → ∞ , i. e. for μ → ∞. In this case, medium #2 acts as a PMC, and it offers Γ = + 1. The PMC has infinite permeability, i.e. μ → ∞. Again, a standing wave is formed in the medium #1, with E_y‐field maximum at the interface; while H_z is zero. The PMC is a hypothetical medium. However, it is realized on the periodically loaded surface as an artificial magnetic conductor (AMC) over a band of frequencies. The interface can also totally reflect the wave if the interface offers either inductive or capacitive impedance. In this case, the interface is a RIS. The periodic surfaces are discussed in chapter 20. These are widely used in the modern microwave and antenna engineering. The PEC, and PMC surfaces, forming the idealized rectangular waveguides, are discussed in chapter 7.

      5.1.3 Transmission Line Model of the Composite Medium

      Both the unbounded medium and transmission line supports the 1D wave propagation. So an unbounded medium could be easily modeled, shown in Fig (5.1b), as a transmission line. The propagation constant of wave on the equivalent transmission line could be treated as identical to that of in the medium. This approach is simple and effective in obtaining the impedance transformation and also impedance matching by using the multilayer dielectric medium. On comparing the wave equations for the E_y (x) and H_z (x), given in equation (4.5.13) of chapter 4, against the voltage and current wave equation (2.1.37) of chapter 2, the following equivalences are observed:

      (5.1.14)

The solution of the voltage and current waves, given in equations (2.1.63) and (2.1.64), can be converted to the solution of E_y and H_z. Finally, the expression of the input impedance of a lossless line of length d, given in equation (2.2.65) of chapter 2, terminated in a load Z_L, helps to write the following expression of the input impedance of a dielectric slab of thickness d; and also the expressions for the reflection coefficient, and transmission coefficient at the interface PQ:

Figure 5.1 Normal incidence of TM‐polarized plane wave at the interface of two media.

(5.1.15)

In expression (5.1.15), medium #2 has an infinite extent. In medium #1, the incident wave is located at a distance d from the interface. The transmission line model is used for the multilayer dielectric medium also [B.1–B.4].

5.2 Oblique Incidence of Plane Waves

      The plane wave of any polarization, obliquely incident at the interface of two dielectric media, can be decomposed to the TE and TM polarizations, so the plane wave of any polarization is a linear combination of the TE and TM polarizations. This section considers the reflection and transmission of both polarizations at the boundary of lossless media. However, losses in the media can also be accounted for [J.1].

      5.2.1 TE (Perpendicular) Polarization Case

Figure (5.2a) shows an oblique incidence of the z‐polarized plane wave, i.e. TE^z polarized ray at the interface P‐O‐Q of two media with different electrical characteristics. The interface PQ is along the y‐axis, and the normal to the interface is along the x‐axis. The plane x‐o‐y containing the normal to the interface, and the rays of the obliquely incident, reflected and refracted waves, i.e. the wavevectors (), is called the plane of incidence. In the present case, the plane of the paper, i.e. the (x − y)‐plane, is the plane of incidence. The incident ray, with the wavevector , strikes the interface at an angle θ₁. It is partly reflected in the medium #1 at an angle and partly refracted in the medium #2 at an angle θ₂. All angles are measured with respect to the normal o‐x. The electric field is perpendicular to the plane of incidence, i.e. in the z‐direction; so it is called the TE‐polarization. It is also known as the perpendicular, i.e. the s‐polarization as the E^inc field is perpendicular to the plane of incidence,

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