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
(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:
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
5.1.2 The Interface of a Dielectric and Perfect Conductor
Medium #2 could be a conducting medium, with the intrinsic impedance,
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 Hz field component can be easily obtained. At the PEC surface, the Ey‐field is zero, and Hz‐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 Ey‐field maximum at the interface; while Hz 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 Ey (x) and Hz (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 Ey and Hz. 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 ZL, 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.
In expression (5.1.15),