Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 171

For the wave propagating only in the x‐direction, equations (4.5.12a) and (4.5.12b) are reduced to the 1D wave equations:

(4.5.13)

Equation (4.5.13a) has the solution, . The time‐harmonic wave propagating in the x‐direction is

(4.5.14)

The field equations in the time‐domain are also written as follows:

(4.5.15)

In the case of a lossy medium, equation (4.5.11) shows that both α and β depend on the loss‐tangent of a medium. For a lossless medium, tan δ = 0, leading to α = 0, and . In the case of low conductivity, i.e. the low‐loss medium, the approximation tan δ ≪ 1, or (σ/ωε₀ε_r) ≪ 1, can be used. In such a medium σ ≪ ωε₀ε_r, the contribution of the conduction current is small as compared to that of the displacement current. Such a medium is a dielectric medium with a small loss. On approximating, the following expression is obtained from equation (4.5.11a):

(4.5.16)

The above equation computes the dielectric loss of a low‐loss dielectric medium. The approximation is used to get an approximate value of β for such medium from equation (4.5.11b):

(4.5.17)

For a low‐loss dielectric medium the dielectric loss, due to tan δ, increases linearly with frequency ω. However, the propagation constant β is dispersionless, giving the frequency‐independent phase velocity. The above approximation can also be carried out in a little different way:

(4.5.18)

In the above equation, is the characteristic (intrinsic) impedance of free space. A low‐loss medium is a mildly dispersive medium, with the frequency‐dependent phase velocity:

(4.5.19)

In the above equation, is the velocity of the EM‐wave in the lossless dielectric medium. The presence of the loss has decreased the phase velocity, i.e. a lossy medium supports the dispersive slow‐wave propagation. The use of α and β from equations (4.5.11a) and (4.5.11b) provide more accurate results. The dispersion in a medium is always associated with loss. This fundamental property is further discussed in chapter 6.