Introduction To Modern Planar Transmission Lines. Anand K. Verma

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      Likewise, for Γ < 0, i.e. for η₂ < η₁, the wave in the medium can be written as follows:

      (5.1.8)

The combined traveling and standing wave description follow a similar condition on a mismatched line given by equation (2.1.89) of chapter 2. Using the phasor – (5.1.6a), the occurs for and occur for , where n = 0,1,2,…. The VSWR in the medium is obtained as follows:

      (5.1.9)

      The power densities in medium #1 (x<0) and #2 (x>0) are obtained using Poynting vector relation:

      (5.1.10)

      The time‐averaged power density is a real part of S^*(x):

      (5.1.11)

      In the case of a lossless composite medium, the power balance is maintained. It is seen by separating the incident, reflected, and transmitted power densities:

(5.1.12)

Equation (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),