Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 236

A lossy DPS medium is characterized by the complex permittivity and complex permeability and , also and . The complex wavenumber for the EM‐wave in a lossy DPS medium is obtained as follows:

(5.5.17)

(5.5.18)

On separating the real and imaginary parts of a complex wavenumber in the DPS medium, k^*DPS = k′ − jk^″, the following expressions are obtained:

(5.5.19)

The DPS medium has . So using the above equations, it is obvious that k^' > 0, k^″ > 0.

The above expressions are also valid for the DNG medium with some modifications. In the case of a DNG medium, the medium parameters are . However, even in the DNG medium, the electric and magnetic losses are positive quantities, i.e. . Using equation (5.5.19), the complex wavenumber in the DNG is given as follows:

(5.5.20)

The complex DPS medium is also described by the complex refractive index:

(5.5.21)

However, in the case of a complex DNG medium, the complex refractive index is n^*DNG = − (n′ + jn″); as Re(n^*DNG) is a negative quantity and Im(n^*DNG) is still a positive quantity. The above discussion is for the propagating waves in the DPS and DNG media. However, in case the waves are nonpropagating (evanescent) in both media, the real part of the wavenumber is an imaginary quantity i.e. k′ = − jα′. The evanescent wave behaves differently in the DPS and DNG media. It is examined below.

The electric fields of the x‐directed propagating and nonpropagating EM‐waves in the unbounded lossy DPS and DNG media could be expressed as follows:

(5.5.22)

The propagating EM‐wave is attenuated while traveling in both the DPS and DNG lossy media k^″ ≠ 0. However, the DPS medium offers a lagging phase, whereas the DNG medium offers a leading phase to the wave traveling in the positive x‐direction. Poynting vector decides the direction of the EM‐wave propagation. The λ_g/2‐line resonator could be designed in the DPS‐DNG composite, with a length λ_g/4 in each medium, without any phase‐shift at the output. The classical λ_g/2‐line resonator, in a DPS medium, has 180° phase at the output. It is further seen from the above equation that the evanescent wave is decaying with distance x while traveling in the DPS medium. The enhancement of amplitude by the DNG medium could be viewed as the step‐up transformer action, whereas it is increasing in amplitude while traveling in the DNG medium. This property is more clearly seen in a lossless medium with k^″ = 0.