Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 170

Maxwell’s coupled equations (4.4.1), in the external source‐free medium , are solved, using the rule of vector algebra, for the electric field intensity by substituting from equation (4.4.1b) to equation (4.4.1a):

(4.5.1)

In the above equation, the identity is used. The above wave equation for the electric field is valid in an isotropic, homogeneous, and lossy medium. In a homogeneous medium, the (μ_r, ε_r) are not a function of position. Further, the medium is taken as charge‐free, i.e. ρ = 0, . Likewise, the following wave equation is obtained for the magnetic field:

(4.5.2)

To get the time‐harmonic field, i.e. , the time differential variable is replaced as follows: ∂/∂t → jω and ∂²/∂t² → − ω². The above wave equations for the time‐harmonic fields are written as,

(4.5.3)

The complex propagation constant γ = α + jβ is defined as follows:

(4.5.4)

For a lossless medium σ = 0, and the propagation constant is a real quantity:

(4.5.5)

In a homogeneous medium, propagation constant β is also expressed as the wavenumber k. In free space, μ_r = ε_r = 1. The velocity of the EM‐wave is equal to the velocity of light (c) in free space:

(4.5.6)

where is the propagation constant, i.e. the wavenumber (k₀) in free space. A lossless material medium is electrically characterized by (ε_r, μ_r). However, it is also characterized by the refractive index . In the case of a dielectric medium, it is . The velocity of the EM‐wave propagation in a medium is

(4.5.7)

For a lossy medium, the complex propagation constant can be further written as:

(4.5.8)

For a lossy dielectric medium, ε_r is defined as a complex quantity:

(4.5.9)

It is like the previous discussion on the complex relative permittivity in a lossy dielectric medium, with the following expressions for the loss‐tangent and propagation constant:

(4.5.10)

In the above equation, the real part of the complex relative permittivity is .

On separating the real and imaginary parts, the attenuation constant (α) and propagation constant (β) are obtained:

(4.5.11)

The wave equation (4.5.3a) and (4.5.3b) for the () fields in a lossy and lossless (α = 0) media are rewritten below:

(4.5.12)

The propagation constant β is also expressed as the wavenumber k of the wavevector . Sometimes in place of the complex propagation constant γ, the complex wavevector k is used as a complex propagation constant, i.e. k^* = β − jα. On using the complex k, the field is written as E₀ e^−jkx = E₀ e^{−j(β − jα)x} = (E₀e^−αx) e^−jβx.