Anand K. Verma - Introduction To Modern Planar Transmission Lines

On solving the above last two equations, the reflection coefficient Γ, and the transmission coefficient τ are computed at the interface:

(5.1.3)

For a lossy medium, the intrinsic impedance is a complex quantity, and reflection and transmission coefficients are also complex quantities. However, the magnitude of the reflection coefficient of a passive medium is always equal to or less than unity, i.e. |Γ| ≤ 1. The magnitude and phase of the reflected and transmitted waves are different from that of the incident waves. For a lossless composite medium, the matching is obtained, i.e. no reflection Γ = 0, for an incident wave at the interface with η 1= η 2. It could be realized in two ways:

(5.1.4)

In equations (5.1.4a,b), the first case is the usual one, as both media are identical. However, the second case is interesting; as the matching is possible even for dissimilar media. It may not be a practical one with natural materials. However, for the artificially engineered metamaterials , both μ rand ε rcan be tuned independently to achieve the matching condition, It is discussed in sub section (5.5.8). The impedance‐matched Huygens' metasurface has also been developed by tuning of electric and magnetic inclusions discussed in subsection (22.5.2) of chapter 22.

Sometimes, the concept of the reflectance (reflectivity) R and transmittance (transmissivity) T are used to show the portion of the reflected and transmitted powers:

(5.1.5)

The total field in medium #1 is a combination of the incident and reflected waves causing interference of waves. The field is uniform in the x ‐direction. However, it creates partly a standing wave, and partly the traveling wave moving in the x ‐direction:

(5.1.6)

For Γ > 0, i.e. for η 2> η 1, the above expression can be decomposed as follows:

(5.1.7)

In equation (5.1.7), is taken.

Likewise, for Γ < 0, i.e. for η 2< η 1, 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 Γ 2+ τ 2= 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].

Medium #2 could be a conducting medium, with the intrinsic impedance, For a PEC medium, σ → ∞ , η 2→ 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 zfield 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 η 2→ ∞ , 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 zis 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.