Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 201

Figure (5.1a) shows an interface of two media #1 and #2. The media are electrically characterized by the primary parameters ε_ri, μ_ri, σ_i; i = 1, 2 and also by the secondary parameters, such as the refractive index n_i, and intrinsic or wave impedance η_i. Initially, both media are considered lossless dielectric media. Next, the second medium is treated as a PEC.

The incident wave in the medium #1, propagating in the x‐direction with propagation constant k₁, is y‐polarized with an electric field component The direction‐x is normal to the interface PQ along the y‐axis; so the incident magnetic field component is +z directed. This is a special case of the TM‐polarized obliquely incident wave, discussed in section (5.2.2). The incident ray strikes at the location O and gets partly reflected and refracted (i.e. transmitted) at O with the field components , and respectively. In the medium #2, the propagation constant is k₂. In general, the k is a wavevector with three components. Figure (5.1a) shows that the direction of propagation is decided by the direction of the Poynting vector. The field components are summarized in equation (5.1.1) on suppressing the time‐harmonic dependence e^jωt:

(5.1.1)

where η₁ and η₂ are the intrinsic impedance of medium #1 and medium #2, respectively. The total tangential components of the electric and magnetic fields in both media are continuous across the interface at x = 0:

(5.1.2)

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 η₁ = η₂. 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_r and ε_r can be tuned independently to achieve the matching condition, It is discussed in subsection (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 η₂ > η₁, 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 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].