Introduction To Modern Planar Transmission Lines. Anand K. Verma

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with interference, and the fields are written from equation (5.2.17) as follows:

(5.2.19)

In equation (5.2.19), time‐harmonic dependence e^jωt is suppressed. At the surface of the perfect conductor x = 0, the field component forms the standing wave. However, the standing wave pattern travels along the interface in the y‐direction.

      5.2.3 Dispersion Diagrams of Refracted Waves in Isotropic and Uniaxial Anisotropic Media

      The dispersion diagrams of the EM-waves in the isotropic and also uniaxial anisotropic medium displaying Snell's law, given in equation (5.2.7), are represented on the (k_z − k_y)‐plane. Snell's law was obtained from the phase matching of the incident and refracted waves across the boundary of two media. The dispersion diagrams in the isotropic and uniaxial anisotropic media are obtained by continuing the process discussed in subsections (4.7.4) and (4.7.5) of chapter 4.

Figure (5.4a) shows the geometrical process of constructing the dispersion diagram showing the refraction of the wave at the interface of the air and dielectric media. The incident wavevector in the air medium (ε_r = 1) subtends an angle of incidence θ_i. The refracted wavevector , in the isotropic dielectric medium with relative permittivity ε_r, subtends an angle of refraction θ_r with respect to the k_z, normal to the interface. Only half of the dispersion circle of each medium is shown. The normal RS intersects the dispersion circle of the isotopic dielectric medium at S. At the interface, the line segments show the phase‐matching across the interface. It is seen that the directions of the phase velocity v_p and group velocity v_g of the refracted wave are the same.

Likewise, Fig (5.4b) shows the geometrical construction of double refraction in the uniaxial anisotropic medium. In this case, the line segments PQ(k₀ sin θ_i) = QR(k_ro sin θ_o = k_re sin θ_e) provide the phase‐matching across the interface. The wavenumbers k_ro and k_re belong to the refracted ordinary and extraordinary waves. The normal drawn from the location R on the dispersion circle to the interface of air-dielectric medium intersects the dispersion circle of the uniaxial anisotropic medium at S₁ for the ordinary waves. It also intersects at the dispersion ellipse at S₂ for the extraordinary waves. The wavevectors OS₁ and OS₂ for the ordinary and extraordinary waves subtend the angle of refraction θ_o and θ_e respectively with k_z‐axis. The first refraction is for the ordinary waves with phase velocity v_po and group velocity v_go in the same direction. The second refraction, for the extraordinary waves, has phase velocity v_pe and group velocity v_ge in different directions. The angles of refraction are different in both cases. The phase and group velocities for both kinds of waves are identical only along the optic axis OO', where the dispersion circle and dispersion ellipse touch each other.

Figure 5.4 Dispersion diagrams of refracted waves.

      

      5.2.4 Wave Impedance and Equivalent Transmission Line Model

      The obliquely incident plane wave is partly reflected and partly transmitted at the interface of two electrically dissimilar media. It helps to think the medium #1 and medium #2 as two dissimilar transmission lines with a junction PQ, corresponding to the interface PQ as shown in Figs (5.2a) and (5.3a) for the TE and TM polarization, respectively. Figures (5.2b) and (5.3b) show the equivalent transmission line networks of the composite media, supporting the oblique incidence of the TE and TM‐polarized plane waves. The source connected to the line #1, with characteristic impedance Z₀, corresponding to the wave impedance of the incident wave in medium #1, is assumed to be located at the left of the junction. The line #2 is of an infinite extent that offers a load Z_L, corresponding to the wave impedance of the transmitted (refracted) wave in medium #2, at the junction. The reflection and transmission coefficients of the incident wave at the physical media interface correspond to the mismatch, causing reflection and transmission, at the junction of two equivalent lines. Our task is to determine the Z₀ and Z_L in terms of the wave impedances of medium #1 and medium #2, respectively, for both the TE and TM polarizations. The correlation between the reflection/transmission coefficient of the TE and TM‐polarized waves at the interface of physical media and reflection/transmission coefficient at the junction of equivalent lines is also be considered.

      Formulation of Transmission Line Model

      At first, the characteristic impedance of equivalent transmission lines, corresponding to wave impedance of both media, is obtained for both the TE and TM‐polarized waves. Next, the relations between reflection/transmission coefficient at the interface of the physical media and reflection/transmission coefficient at the junction of equivalent lines are obtained.

      Correspondence between Wave Impedance and Characteristic Impedance

      The wave impedance of the incident, and transmitted TE waves in the medium #1 and medium #2, as shown in Fig (5.2a), with respect to the direction of propagations k₁ and k₂ are given below:

      (5.2.20)

However, the interface in the (y − z)‐plane views both the above‐given wave impedances differently due to the oblique incidence. The left (x = 0⁻) and right (x = 0⁺) side faces view the following x‐directed wave impedances, and , due to the (E_z, H_y) field components:

      (5.2.21)

      

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